TORIC GEOMETRY

Math 7800 - Fall 2002

Class time: Tuesdays and Thursdays from 1:00pm -- 2:20pm in LCB222.

This course is aimed at students of algebraic geometry, but geometers of any flavour may find the subject interesting, particularly given the applications to mirror symmetry. Basic knowledge of algebraic geometry is required. Ambitious students participating in the introductory algebraic geometry class Math 6130 should have no trouble studying both topics.

• Course announcement.

1. Lectures 1 and 2: Introduction to toric geometry.
   The first two lectures give a brief overview of the content of the course, focusing on some simple examples.

2. Lectures 3,4 and 5: The construction of toric varieties.
   Fans are defined and the corresponding toric variety is constructed.

3. Lectures 6 and 7: The torus action on a toric variety.
   A toric variety is decomposed into torus orbits, the E-polynomial computes Hodge-Deligne numbers and the orbit closures are constructed.

4. Lectures 8, 9 and 10: Affine toric surfaces and warm-up to resolutions.
   Classification of toric surfaces, Jung-Hirzebruch resolution, discrepancy calculations and description of minimal resolution as G-Hilb.

5. Lectures 11 and 12: Toric morphisms, smoothness and resolution of singularities.
   Normality, criteria for smoothness and compactness, proper morphisms and toric resolution of singularities.

6. Lectures 13, 14, 15 and 16: Toric orbifolds and 3-fold singularities.
   Abelian quotient singularities, canonical and terminal singularities with a focus on 1/r(1,a,r-a) and Abelian quotients by subgroups of SL(3,C).

7. Lectures 17,18 and 19: Birational geometry of toric varieties
   Classification of complete nonsingular toric surfaces, strong and weak factorisation of birational maps between toric surfaces, weak factorisation for toric 3-folds.

8. Lectures 20 and 21: Divisors and line bundles
   Divisors, torus-invariant divisors and support functions, the canonical class of toric varieties.

9. Lecture 22: Projectivity of toric varieties
   Toric surfaces, a complete nonprojective toric variety, toric varieties from polytopes.

10. Lectures 23, 24,25 and 26: Minimal models of toric varieties
    Intersection theory, intro to the MMP, toric cone theorem, toric contraction theorem and the toric flip theorem.

1. T.Oda Convex bodies and algebraic geometry, Springer Verlag Survey in Mathematics 15 (1988).
2. W. Fulton. Introduction to Toric Varieties, Princeton University Press 1993.